Average-case analysis of a greedy algorithm for the 0/1 knapsack problem

We consider the average-case performance of a well-known approximation algorithm for the 0/1 knapsack problem, the Decreasing Density Greedy (DDG) algorithm. Let Un = {u1, . . . , un} be a set of n items, with each item ui having a size si and a profit pi, and Kn be the capacity of the knapsack. Given an instance of the 0/1 knapsack problem, let PL denote the total profit of an optimal solution of the linear version of the problem (i.e., a fraction of an item can be packed in the knapsack) and PDDG denote the total profit of the solution obtained by the DDG algorithm. Assuming that Un is a random sample from the uniform distribution over (0, 1]2 and Kn = σn for some constant 0 < σ < 1/2, we show that √ n(PL − PDDG) converges in distribution.